Question

What is the discriminant of `-x^2+10x-56=-4x-7`?

Answer 1

For this quadratic, `Delta = 0`.

Explanation:

In order to determine the determinant of this quadratic equation, you must first get it to quadratic form, which is

`ax^2 + bx + c = 0`

For this general form, the determinant is equal to

`Delta = b^2 - 4 * a * c`

So, to get your equation to mthis form, add `4x + 7` to both sides of the equation

`-x^2 + 10x - 56 + (4x + 7) = -color(red)(cancel(color(black)(4x))) - color(red)(cancel(color(black)(-7))) + color(red)(cancel(color(black)(4x))) + color(red)(cancel(color(black)(7)))`

`-x^2 + 14x - 49 = 0`

Now identify what the values for `a`, `b`, and `c` are. In your case,

`{(a = -1), (b=14), (c=-49) :}`

This means that the discriminant will be equal to

`Delta = 14^2 - 4 * (-1) * (-49)`

`Delta = 196 - 196 = color(green)(0)`

This means that your equation has only one real root

`x_(1,2) = (-b +- sqrt(Delta))/(2a)`

`x = (-b +- sqrt(0))/(2a) = color(blue)(-b/(2a))`

In your case, this solution is

`x = (-14)/(2 * (-1)) = 7`